If you were to do that, the Goldilocks zone would move inward as the Sun’s energy output dropped. So all the life on Earth would still die because the Earth would freeze.

Still, since you have the tech to remove mass from a star, you would also surely have the ability to move the earth inward to keep it in the Goldilocks zone. But that still might not work. Being so much closer to the Sun, the Earth’s magnetic field and atmosphere might not be sufficient to block flares and CMEs.